Polar Forms, p-Values, and the Core

Васильев, В. А.

doi:10.1007/978-3-642-57592-1_30

Cited by 21 publications

(50 citation statements)

References 5 publications

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“…The following assertion holds for the free-time problem [5,6]: (4), then there exists a vector a = (a 0 , a 1 ) and a function (t), t 2 [0, T ], that satisfy the conditions…”

Section: Maximum-principle Boundary-value Problem Summary Of Resultsmentioning

confidence: 97%

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Optimal Control Problem with an Integral Functional of an Exponential Control Function

et al. 2014

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“…The following assertion holds for the free-time problem [5,6]: (4), then there exists a vector a = (a 0 , a 1 ) and a function (t), t 2 [0, T ], that satisfy the conditions…”

Section: Maximum-principle Boundary-value Problem Summary Of Resultsmentioning

confidence: 97%

“…Problem (4) has an optimal control in the class of Lebesgue-measurable functions [4]. To investigate the optimal control problem (4), we apply the necessary optimality conditions of parametric processes in the form of Pontryagin's maximum principle for problems with free time [5,6].…”

Section: Statement Of the Problemmentioning

confidence: 99%

Optimal Control Problem with an Integral Functional of an Exponential Control Function

et al. 2014

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“…Thus, we have established property (19) of the difference operator A. Let us pass to the stability analysis of difference scheme (16), (17) with respect to initial data.…”

Section: Difference Schemementioning

confidence: 99%

“…To obtain the approximate solution of the optimal control problem (4)-(9), one can use gradient methods [17,19] and the difference scheme (16), (17) proposed for the numerical solution of the direct problem (4)- (7). The double-layer difference scheme (16), (17) can also be applied directly to the numerical solution of the adjoint problem (30)-(33).…”

mentioning

confidence: 99%

Investigation of Wave Processes in Inhomogeneous Media with Imperfect Contact Conditions

Gladky

2016

Cybern Syst Anal

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The author considers the problem of numerical modeling and optimization of wave processes in inhomogeneous domains with imperfect contact conditions on the basis of Schr&& odinger parabolic wave equation. The optimality criterion is formulated. The differential properties of the optimization problem are investigated. The numerical method is proposed for modeling and optimization of acoustic fields in inhomogeneous domains.Numerical and numerical-analytical methods of mathematical modeling of sound wave propagation have been developed in the last decades in ocean acoustics to solve distant sounding and acoustic monitoring problems [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Of considerable interest, first of all for geophysics and ocean acoustics, are problems related to formation of acoustic fields with prescribed properties and analysis of the propagation of sound waves in inhomogeneous waveguides with regard for thin inclusions. The mathematical difficulties that arise in solving such problems can be overcome by approximating the Helmholtz wave equation with parabolic equations [3-7], which necessitates the development of numerical methods to solve direct and extremum problems for Schr&& odinger-type equations with complex-valued non-self-adjoint operator.In the paper, we will analyze numerical modeling and optimization of acoustic fields on the basis of parabolic wave equation in an axisymmetric inhomogeneous double-layer waveguide with piecewise continuous (piecewise constant) acoustic parameters and imperfect-contact conditions, which allows taking into account thin inclusion in the waveguide. We will propose a numerical method to solve the direct and optimization problems, investigate the differential properties of the quality functional, and analyze the stability of difference scheme for the direct and adjoint boundary-value problems with complex-valued non-self-adjoint operator.

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“…An approximate solution of optimal control problem (9)-(13) can be obtained using gradient methods [16,17]. Therefore, in such problems, the gradient of the functional being minimized should first be computed and necessary optimality conditions should be formulated.…”

Section: Investigation Of the Extremal Problemmentioning

confidence: 99%

Numerical modeling and optimization of one-way wave processes in inhomogeneous media

Gladky

Skopetsky

2010

Cybern Syst Anal

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The problem of numerical modeling and optimization of wave processes on the basis of a parabolic Schrodinger-type wave equation with a complex non-self-adjoint operator is considered. An optimality criterion is formulated. Properties of an extremal problem are investigated. A numerical method is proposed for the modeling and optimization of acoustic fields in inhomogeneous domains with piecewise continuous parameters.The urgency of development of methods for the mathematical modeling of processes of propagation of acoustic waves in inhomogeneous underwater waveguides is mainly explained by needs for remote sensing and acoustic monitoring of regions of the World Ocean [1-4]. Sound waves are used practically in all types of signalling, communication, location, and remote investigations of the body of water and the bottom of the Ocean. Of significant interest are also questions of formation of acoustic fields with prescribed properties and investigations of distinctive features of propagation of sound waves in inhomogeneous waveguides with allowance for absorption in a medium.From the mathematical viewpoint, the calculation of harmonic sound fields is reduced to the solution of boundary problems for Helmholtz's wave elliptic equation with a complex non-self-adjoint operator in unbounded inhomogeneous domains. At present, many numerical-analytical methods and computational algorithms are developed to solve direct or extremal problems for Helmholtz's equation in homogeneous and nonuniform stratified media [1-2, 5-9]. Difficulties of mathematical investigation of such problems can be overcome using the methodology of approximation of the Helmholtz equation by parabolic Schr && odinger-type equations [3][4][5][10][11][12][13][14]. This allows one to reduce the solution of boundary problems to the solution of the Cauchy problem for wave equations of parabolic type with a complex non-self-adjoint operator, which leads to the necessity of development of efficient numerical methods for the solution of the corresponding approximation problems.In the present article, questions of numerical modeling and optimization of acoustic processes are investigated on the basis of a parabolic wave equation in an inhomogeneous underwater waveguide with piecewise continuous acoustic parameters and with allowance for absorption. A numerical method is proposed for the solution of the direct problem and optimization problems, differential properties of a quality functional are studied, and the stability of the difference scheme is investigated for the direct and conjugate boundary problems with a complex non-self-adjoint operator.

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Polar Forms, p-Values, and the Core

Cited by 21 publications

References 5 publications

Optimal Control Problem with an Integral Functional of an Exponential Control Function

Optimal Control Problem with an Integral Functional of an Exponential Control Function

Investigation of Wave Processes in Inhomogeneous Media with Imperfect Contact Conditions

Numerical modeling and optimization of one-way wave processes in inhomogeneous media

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